Parabolic Procedure for Flows in Ducts with Arbitrary Cross Sections

Abstract

A computer program has been developed to predict three-dimensional compressible viscous flows in ducts with arbitrary cross-sectional shapes. A curvilinear boundary-fitted coordinate system is used to simplify boundary conditions. The parabolized Navier-Stokes equations are transformed, and the solution is marched down the duct using an iterative ADI procedure. Computed results are compared with test data for laminar test cases in square and round ducts. A two-equation turbulence model is demonstrated for a developing turbulent pipe flow. A comparison between predicted results and test data is presented for a turbulent diffuser flow with a rectanguiar-to-round transition in cross-sectional geometry.

Review—Computational Methods for Internal Flows With Emphasis on Turbomachinery

“…The cross flow equations retain a separate pressure correction p c which is permitted to vary over the cross section. This procedure is employed by Patankar and Spalding [108], Briley [109], Ghia et al [110], Roberts and Forester [111], and Briley and McDonald [112].…”

“…13 Computed and measured axial velocity profiles after 77.5 deg turning in circular arc square duct-after Kreskovsky [118] the pressure correction. Roberts and Forester [111] work directly with the divergence of the transverse momentum equations which gives a 2D elliptic equation with a source term related to the non-satisfaction of local continuity. This equation is solved iteratively with the momentum equations until continuity is satisfied.…”

Computational Procedure for Developing Turbulent Flow and Heat Transfer in a Square Duct

“…The calculations were started by using the explicit method for approximately two hydraulic diameters and then shifted to the DuFort-Frankel method. The explicit method always broke down at large values of x, not because of numerical instabilities, but because an inflection point developed in the U velocity profile: This was apparently caused by an inadequate transverse momentum advection and was cured only by refining the mesh at selected x values, with a consequent reduction in A x (see [21] for a discussion of this point). On the other hand, the DuFort-Frankel method could not be used from the entrance because of instabilities.…”

“…The mesh used is illustrated in Fig. 3 The coefficients were evaluated at the previous axial location i and were not reevaluated in the iterative solution for Vi+ since the effect of such a reevaluation was found to be small (any errors created by this linearization were found to be self-correcting at the next axial station, which is similar to the correction method employed in [21]). …”

“…The DuFortFrankel technique has some computational advantages but also suffers from excessive dissipation under some conditions. Roberts and Forester [21] give an excellent summary of the various techniques and their limitations.…”